Question

in how many ways can the letters of the word multiple be added to the changing the order of the vowels

Answer 1

 the question correctly, the word multiple has 8 letters, and you can figure out how many combinations (the full set) with 8 letters simply enough, but if you restrict the problem to current vowel order, which is "u comes before i" then you can simply subtract off those cases from the full set in which "i comes before u."

There is probably a way to pre-condition that restraint, but I don't know it offhand. I'll let you do that because this sounds like a stat-set problem, and one never learns by reporting another's answer.